Hi! This is my first response to anything here, so please be kind…

OK, about scales. First of all, sound frequency is a continuum, and the pitches we pick out are discrete points in that continuum. How many tones there are in an octave will depend on how closely or widely spaced those pitches are. So when we talk about a pentatonic scale we mean one with five pitches to the octave. The sixth pitch would be an octave above the first one. To change a five-note scale to a seven-note one you have to divide the octave differently.

To hear the pentatonic scale, just play only the black keys on a piano. Several people have mentioned that Chinese music uses this scale, but it has also been used all over the world, even in Western music. Lots of hymns and folktunes are pentatonic: e.g. Amazing Grace.

There is real physics behind the choice of pitches for a scale, which is why the same scales show up in different cultures. Different pitches have different frequencies of sound (i.e. how fast or slowly whatever is producing the sound is vibrating), and these frequencies are related to one another. In the middle ages, Music was taught in the universities as a part of mathematics – it was the science of Ratio rather than what we would consider music class to be today.

In the middle ages in France they would demonstrate the ratios with strings (it still works, btw – I’ve tried it in class). Take a string and divide it in half (producing the ratio 2:1). Half the string will vibrate exactly twice as fast as the whole string, and produce a pitch exactly one octave above that of the whole string. You can generate all the pitches in the scale this way, dividing the string into three and then four parts:

3:2 = the interval of a fifth (e.g. C-G on a piano)

4:3 = the interval of a fourth (e.g. C-F or G-C etc)

Dividing the string into 9 parts, you can generate a single tone (e.g. from C to D on the piano), with the sound ratio of 9:8. The smaller the numbers in a ratio, the better the two pitches will sound to us (music rather than noise).

Using different pitches as the basis for generating new pitches, you can find all seven pitches of Western scales this way. The tradition is that the philosopher Pythagoras (6th century BC) discovered this method of determining the pitches of a scale – the kind of tuning this method produces is called “pythagorean” to this day. And this is why Western scales have seven pitches, with the 8th an octave above the first: we’ve been teaching ourselves to divide the octave this way for over two thousand years.

Someone has already mentioned the American composer Harry Partch. He was fascinated with the history of tuning and intonation. His book *Genesis of a Music* (1949, 2nd ed. Da Capo Press, 1974) has a chapter on the history of tuning systems that is the best and most consise discussion of this topic that I know.

One last thing: nobody asked it, but I can tell you where the “do re mi” syllables come from. Guido of Arezzo (an Italian musician/teacher who lived c.1000-1050) taught his students to sing a major scale using a gregorian chant hymn to St. John the Baptist called “Ut queant laxis”. Guido had noticed that each phrase of the hymn began on a different scale pitch, starting on C and moving up. He used the syllables that began each phrase as the name for that pitch:

UT queant laxix

REsonare fibris

MIra gestorum

FAmuli tuorum,

SOLve polluti

LAbii reatum, Sancte Joannes.

[“That thy servants may freely proclaim the wonders of thy deeds, absolve the sins of their unclean lips, O holy John”]

“Ut” (pronounced Oot) was changed to “Do” (pronounced Doh) in the 20th century. But I have no idea who did it, or why.